FOM: determinate truth values, coherent pragmatism

[Joe Shipman]

On September 4th, Friedman wrote:

<< 4. I also see the possibility that the general math community may be
more
 comfortable with "there is an atomless probability measure on all
subsets
 of the unit interval" than with large cardinal axioms, in that this
 involves objects that are far less abnormal than a large cardinal. Of
 course, this particular one and a measurable cardinal are known to be
 equivalent for normal mathematical purposes. (Actually this equvalence
 passes through some sophisticated set theory that is not "convenient"
for
 mathematicians, and so there is a need to put things in a form that
 mathematicians will find easy to use). Of course, a byproduct of this
 probability measure is that CH is very badly false. But, when viewed as
an
 approach to CH, this is seriously at odds with the approach currently
taken
 by the set theorists. After all, if they like it, then the CH would
have
 been viewed as being settled long long ago by them. They reject this
 probability measure as an axiom candidate. However, from the coherent
 pragmatism that I see in the math community, they may well accept this
when
 shown how to use this in an easy and coherent and effective way for a
 sufficient body of attractive normal mathematics. This would be where
the
 set theorists' approach  via some notion of truth and the
mathematicians
 approach via coherent pragmatism would clash. >>

On September 5th, I replied:

<<Harvey, I agree with most of your latest post, but I am not sure why
you are
making a distinction between set theorists and other mathematicians
regarding
approach to new axioms:  it seems to me that the set theorists are also
using
"coherent pragmatism" (though they may like to talk in terms of truth).
I
have certainly not seen any satisfactory arguments from set theorists
why the
axiom of an atomless measure on the continuum is FALSE; though I have
seen
arguments that alternatives to this axiom (such as Martin's axiom) are
USEFUL, I have not seen any to persuade me that those alternatives are
"true".

Can any set theorists reading this who take a realist view and are of
the
opinion that the "atomless measure" axiom is actually false (rather than

unprofitable to study) please explain the reasons for this opinion?>>

The only response from a set theorist was Steel's on September 6th:

<<It would be most useful to have a broadest point of view about sets
accepted by all. If different points of view arise, it will be of
immediate practical importance to put them together appropriately, so
that
we can continue to use each other's work. I think set theorists are
engaged in uncovering such a broadest point of view, and deciding the CH

is the next fundamental problem along this (never-ending) road. (The
existence of a real valued measure, which is a statement of 4th order
arithmetic, is significantly further off.) >>

I replied on September 17th:

<<Well, if you decide CH positively, you decide the existence of a
real-valued
measure negatively.

Harvey made the excellent point that propositions with plausible
alternatives
are in a very different class than axioms whose negation doesn't get you

anywhere.  Thus "a measurable cardinal exists" is a very different
animal
from "a measurable cardinal is consistent" because V=L is a coherent
alternative to the former but no plausible view of sets seems to
contradict
the latter.  It seems to me that when considering CH vs RVM, which are
real
alternatives to each other and can't both be true, it won't be enough to
seek
a "broadest point of view".>>

I then repeated my query above on RVM, but no further responses from set
theorists were noted.

Therefore, I draw the conclusion that Harvey is wrong, and that set
theorists do NOT differ from "ordinary mathematicians" in their approach
to new axioms, because if they WERE in fact motivated "via some notion
of truth" rather than "the mathematicians approach via coherent
pragmatism", someone would have been able to explain their rejection of
RVM by discussing reasons it might actually be false.

-- Joe Shipman